用半定规划证明Costa熵幂不等式和微分熵的高阶不等式
Prove Costa's Entropy Power Inequality and High Order Inequality for Differential Entropy with Semidefinite Programming
摘要：
Costa熵幂不等式是Shannon熵幂不等式的一个重要推广.与Costa的熵幂不等式和McKean于1966年提出的一个猜想有关，Cheng-Geng最近证明了$D(m,n): (-1)^{m+1}(\partial^m/\partial^m t)H(X_t)\ge0$，其中$X_t$是Costa熵幂不等式中的$n$维随机变量，$H(X_t)$是$X_t$的微分熵. Costa证明了$D(1,n)$和$D(2,n)$是Costa熵幂不等式的结果。Cheng-Geng证明了$D(3,1)$和$D(4,1)$。本文给出了半定规划下$D(m,n)$和Costa熵权不等式的一个系统的证明过程。利用基于此过程的软件包，我们证明了$D(3,n)$对于$n=2,3,4$，并给出了Costa熵幂不等式的一个新的证明.我们还表明，与目前已知的限制，$D(5,1)$和$D(4,2)$不能证明的过程。
英文摘要：
Costa's entropy power inequality is an important generalization of Shannon's entropy power inequality. Related with Costa's entropy power inequality and a conjecture proposed by McKean in 1966, Cheng-Geng recently conjectured that $D(m,n): (-1)^{m+1}(\partial^m/\partial^m t)H(X_t)\ge0$, where $X_t$ is the $n$-dimensional random variable in Costa's entropy power inequality and $H(X_t)$ the differential entropy of $X_t$. $D(1,n)$ and $D(2,n)$ were proved by Costa as consequences of Costa's entropy power inequality. Cheng-Geng proved $D(3,1)$ and $D(4,1)$. In this paper, we propose a systematical procedure to prove $D(m,n)$ and Costa's entropy power inequality based on semidefinite programming. Using software packages based on this procedure, we prove $D(3,n)$ for $n=2,3,4$ and give a new proof for Costa's entropy power inequality. We also show that with the currently known constraints, $D(5,1)$ and $D(4,2)$ cannot be proved with the procedure.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
